On affine Lusztig varieties

Affine Lusztig varieties encode the orbital integrals of Iwahori–Hecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. We establish a close relationship between affine Lusztig varieties and affine Deligne–Lusztig varieties. Consequently, we give an explicit nonemptiness pattern and dimension formula for affine Lusztig varieties in most cases.